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NCERT's chapter on populations names one equation after two people: "Verhulst-Pearl Logistic Growth". Most students learn the formula and the S-shaped curve without knowing who Verhulst was or why his idea sat unused for 80 years. This post tells that story, then works through the logistic equation with numbers, including the result that growth is fastest at half the carrying capacity.
Who was Pierre Verhulst?
The MacTutor biography from the University of St Andrews gives his dates: born 28 October 1804 in Brussels, died there on 15 February 1849. At the Athenaeum of Brussels he was taught mathematics by Adolphe Quetelet, who became his friend and adviser and who was himself interested in population. Verhulst was elected president of the Belgian Academy of Sciences in 1848, the year before he died.
What was his idea?
The belief before Quetelet and Verhulst, says MacTutor, was that a growing population followed a geometric progression, as Thomas Malthus had argued. In his 1838 paper, *Notice sur la loi que la population suit dans son accroissement*, Verhulst accepted Malthus's doubling only if one ignored "the increasing difficulty to find food". Because a country's size and fertility are limited, he wrote, the population "gets closer and closer to a steady state". He proposed a differential equation now called the logistic equation.
In a communication of 30 November 1844, published in 1845, he named its solution the "logistic" curve. He never explained the name. In that paper he also estimated an upper limit for Belgium's population of 9,400,000. A third paper followed in 1847, in which he criticised his own model.
For the man he was answering, see our post on Thomas Malthus and his essay on population.
Why is it called Verhulst-Pearl growth?
MacTutor says Verhulst's equation was then largely ignored until Raymond Pearl and Lowell Reed's work in 1920. Their paper in PNAS fitted a curve of this type to United States census counts from 1790 to 1910. They wrote that its upper asymptote "has the value 197,274,000 roughly", about twice the population of the time, and called their numbers "only a rough first approximation".
The real population went far past that. The 2020 Census counted 331,449,281 residents, about 1.68 times Pearl and Reed's ceiling. Even allowing for Alaska and Hawaii, which were not in their calculation, the ceiling was well short. The lesson is not that the S-shape is wrong. A fitted K describes conditions at the time of fitting, and when conditions change, the limit can move.
A worked example with the logistic equation
NCERT's Class 12 chapter on organisms and populations writes the model as dN/dt = rN(K − N)/K, where N is population density, r the intrinsic rate of natural increase and K the carrying capacity.
Take r = 0.5 per year and K = 1000, practice values for a hypothetical pond population.
At N = 200: dN/dt = 0.5 × 200 × (1000 − 200)/1000 = 100 × 0.8 = 80 per year.
At N = 500: dN/dt = 0.5 × 500 × 0.5 = 125 per year.
At N = 900: dN/dt = 0.5 × 900 × 0.1 = 45 per year.
Growth is slow at the start, fastest in the middle and slow again near K. Where exactly is the peak? Write dN/dt = r(N − N²/K). Differentiate with respect to N: r(1 − 2N/K), which is zero when N = K/2. Substituting, the maximum rate is r × (K/2) × (1/2) = rK/4 = 0.5 × 1000/4 = 125, matching step 2.
If you know integration, the solution is N(t) = K / (1 + ((K − N₀)/N₀)e^(−rt)). Start with N₀ = 100:
(K − N₀)/N₀ = 900/100 = 9.
At t = 4 years: e^(−2) ≈ 0.1353, so 9 × 0.1353 ≈ 1.218 and N ≈ 1000/2.218 ≈ 451.
Time to reach K/2 = 500: set 9e^(−0.5t) = 1, so t = ln 9 / 0.5 ≈ 2.197/0.5 ≈ 4.39 years.
Common mistakes:
Writing (N − K)/K. That makes growth negative below K. The bracket is (K − N)/K, the fraction of room left.
Putting the fastest growth at K. At K the bracket is zero, so growth stops. The steepest point of the S is at K/2.
Mixing up r and the actual growth rate. r is the rate with unlimited room; the actual per-head rate is r(K − N)/K, which shrinks as N rises.
To practise the calculus side, our guide to differential equations for JEE covers separating variables.
What can a student take from his way of working?
Fix a model, don't just reject it. Verhulst kept Malthus's growth and added one term for limits. Small, well-argued changes are often the strongest.
Put a number on it. Verhulst turned his model into a concrete limit for Belgium, and Pearl and Reed later fitted census counts. A model that gives numbers can be checked.
Criticise your own work. His 1847 memoir questioned his own model.
If the S-curve is easier to see than read, the Fermi Simulation Lab has interactive simulations for ideas easier to see than read, so you can watch how r and K change the shape.
Try it now: with r = 0.4 and K = 2000, find the population at which growth is fastest and the maximum rate, then check your answers using rK/4.
